Question: Find the distance between the foci of the hyperbola $x^2 - 6x - 4y^2 - 8y = 27.$
Explanation: Completing the square in $x$ and $y,$ we get
\[(x - 3)^2 - 4(y + 1)^2 = 32.\]Then
\[\frac{(x - 3)^2}{32} - \frac{(y + 1)^2}{8} = 1.\]We see that $a^2 = 32$ and $b^2 = 8,$ so $c^2 = a^2 + b^2 = 40,$ and $c = 2 \sqrt{10}.$  Therefore, the distance between the foci is $2c = \boxed{4 \sqrt{10}}.$